Abstract
In this paper, we derive the cumulative distribution functions (CDF) and probability density functions (PDF) of the ratio and product of two independent Weibull and Lindley random variables. The moment generating functions (MGF) and the k-moment are driven from the ratio and product cases. In these derivations, we use some special functions, for instance, generalized hypergeometric functions, confluent hypergeometric functions, and the parabolic cylinder functions. Finally, we draw the PDF and CDF in many values of the parameters.
Highlights
Equation (24) is probability density functions (PDF) of z-generalized hypergeometric function
Weibull distribution X ∼ W(a, b) with shape parameter a > 0 and scale parameter b > 0, the PDF and cumulative distribution functions (CDF) are defined as f(x)
The plots of CDF and PDF are drawn. e m-moment, mean, and variance are calculated. e CDF and PDF are derived in two formulas with respect to each one of them, first formula by confluent hypergeometric function and another formula by generalized hypergeometric function
Summary
Equation (24) is PDF of z-generalized hypergeometric function. We put equation (23) in equation (13) to get the following: E. Weibull distribution X ∼ W(a, b) with shape parameter a > 0 and scale parameter b > 0, the PDF and CDF are defined as f(x) Lindley distribution Y ∼ L(c) with shape parameter c > 0, the PDF and CDF are given as c2 f(y) c + 1 (y + 1)exp[− cy]. The PDF, equation (1), and CDF, equation (2), become, respectively, x2 f(x) b2 x exp−
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